Smooth solutions to the nonlinear wave equation can blow up on Cantor sets

Abstract

We construct $C^\infty$ solutions to the one-dimensional nonlinear wave
equation $$ u_{tt} - u_{xx} - \tfrac{2(p+2)}{p^2} |u|^p u=0 \quad \text{with}
\quad p>0 $$ that blow up on any prescribed uniformly space-like $C^\infty$
hypersurface. As a corollary, we show that smooth solutions can blow up (at the
first instant) on an arbitrary compact set.
We also construct solutions that blow up on general space-like $C^k$
hypersurfaces, but only when $4/p$ is not an integer and $k > (3p+4)/p$.

Full text

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arXiv:1103.5257v1 [math.AP] 27 Mar 2011
SMOOTH SOLUTIONS TO THE NONLINEAR WAVE EQUATION
CAN BLOW UP ON CANTOR SETS
ROWAN KILLIP AND MONICA VIS ¸AN
Abstract. We construct C∞solutions to the one-dimensional nonlinear wave
equation
utt− uxx−2(p+2)
p2
that blow up on any prescribed uniformly space-like C∞hypersurface. As a
corollary, we show that smooth solutions can blow up (at the first instant) on
an arbitrary compact set.
We also construct solutions that blow up on general space-like Ckhyper-
surfaces, but only when 4/p is not an integer and k > (3p + 4)/p.
|u|pu = 0withp > 0
1. Introduction
In this paper we consider the one-dimensional focusing nonlinear wave equation
utt− uxx−2(p+2)
p2
|u|pu = 0(1.1)
for general powers p > 0. The coefficient of the nonlinearity can be changed via
the rescaling uλ(t,x) := u(λt,λx). However, this particular choice simplifies many
of the formulae that follow.
Even for smooth initial data, solutions to (1.1) can blow up in finite time. The
simplest example of this is
u(t,x) = (T − t)−2/p
defined for t < Tandx ∈ R.(1.2)
(Note the benefit of choosing the coefficient 2(p + 2)/p2in (1.1).)
The equation (1.1) is relativistically invariant. This blurs the distinction between
space and time, as well as implying finite speed of propagation. As a result, it is
natural to consider the maximal space-time region on which the solution is defined.
Moreover, this region must take the form
{(t,x) : ˜ σ(x) < t < σ(x)},
where either σ(x) ≡ ∞ or σ is a 1-Lipschitz function, that is,
sup
x,y
|σ(x) − σ(y)|
|x − y|
≤ 1,
and either ˜ σ(x) ≡ −∞ or ˜ σ is a 1-Lipschitz function. (The number 1 represents
the speed of light in our model.) A point t = σ(x) on the (forward) blowup surface
is called non-characteristic if
|σ(x) − σ(y)|
|x − y|
otherwise, it is called characteristic. The surface {t = σ(x)} is called space-like if
it is everywhere non-characteristic.
limsup
y→x
< 1;
1

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2ROWAN KILLIP AND MONICA VIS ¸AN
In this paper we discuss the question of which surfaces can occur as blowup
surfaces. In the papers [3, 4], Caffarelli and Friedman showed that σ must be C1at
non-characteristic points for a closely related model (they also obtain conditional
results in higher dimensions). More recently, Merle and Zaag considered precisely
the model (1.1) in a series of papers; see [10, 11] and the references therein. They
have shown that the characteristic points form a discrete set and that σ is C1away
from these points. (The also obtain a wealth of information about how the solution
behaves near the blowup surface.) In [12], Nouaili extended their methods to show
that away from the characteristic points, σ is C1,αfor some (unspecified) H¨ older
exponent α > 0.
In the other direction, one may seek to show that certain rich classes of functions
σ do indeed arise as blowup surfaces. The only works we know of that address
this type of question are the papers [8, 9] of Kichenassamy and Littman. These
authors consider arbitrary spatial dimension with p ∈ N and show that any analytic
space-like surface can occur as a blowup surface. In his lecture notes on the subject,
Alinhac explicitly asks about the possibility of extending the work of Kichenassamy
and Littman to the C∞case; see [2, p. 9]. In this paper, we work only in one
dimension and show that general C∞space-like surfaces can indeed occur as blowup
surfaces. Our methods apply also in the case of finite regularity; however, for
technical reasons, the argument becomes messier if 4/p is an integer, so we are
omitting that case.
Theorem 1.1 (Regular solutions that blow up on a space-like Ckhypersurface).
Suppose either (i) p > 0 and k = ∞, or (ii)
Σ = {t = σ(x)} be a Ckuniformly space-like hypersurface (i.e., supx|σ′(x)| < 1).
Then there exist a nearby space-like hypersurface Σ0:= {t = σ0(x)} with σ0(x) <
σ(x) and a solution u ∈ Cm
that blows up on Σ in the sense that
4
p?∈ N and 3 +4
p< k ∈ N ∪ {∞}. Let
t,x({σ0(x) ≤ t < σ(x)}) to (1.1), with m = ⌊k−3−4/p⌋,
lim
tրσ(x)
?σ(x) − t?2
pu(t,x) =?1 − |σ′(x)|2?1
p
(1.3)
for all x ∈ R.
Remarks. 1. The floor notation, ⌊x⌋, denotes the greatest integer less than or equal
to x. Similarly, the ceiling ⌈x⌉ is the least integer greater than or equal to x.
2. The blowup rate in (1.3) is Lorentz invariant: to this degree of accuracy,
u(t,x) agrees with the Lorentz boost of the solution (1.2) that blows up on the
tangent line to Σ passing through the point t = σ(x).
3. The space-like hypersurface {t = σ0(x)} may follow the blowup surface {t =
σ(x)} so closely that our solution u is not defined globally in space at any one
particular time. (This issue also appears in results of [9].)
4. The condition k > 3+4
parises as a natural requirement for our method. This
is discussed at some length in Section 5.
Our initial interest in treating the case of smooth (as opposed to analytic) blowup
surfaces was to address the question alluded to in the title: Can solutions to nonlin-
ear wave equations blow up on Cantor-like sets? By this we mean that at the time
when the first singularity occurs, can this singularity arise simultaneously across a
Cantor-like set? Note that if σ is an analytic function, then {x : σ(x) = T} must be
discrete (or all of R); however, given an arbitrary compact set E, it is not difficult

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BLOWUP ON SMOOTH HYPERSURFACES FOR NLW3
to construct a smooth function for which E is a level set. In this way, Theorem 1.1
allows us to construct a solution to (1.1) that blows up on an arbitrary compact set.
Moreover, we are able to control the time of existence sufficiently well that we can
guarantee that the solution admits global (in space) Cauchy data (cf. Remark 3
above). The precise statement is as follows:
Corollary 1.2 (Blowup on arbitrary compact sets). Fix p > 0. Given a compact
set E ⊂ R, there exist C∞(R) initial data (u(0),ut(0)) and a smooth solution to
(1.1) with this initial data that blows up at a time 0 < T < ∞ precisely on the
set E, which is to say,
• u(t,x) converges to a smooth function of x ∈ R \ E when t → T and
• limt→Tu(t,x) = ∞ for all x ∈ E.
The proof of Theorem 1.1 proceeds by constructing a parametrix ˜ u for the desired
solution and then using contraction mapping arguments to correct it to an actual
solution u to (1.1). Due to the singularity of the solution we are seeking, the
equation obeyed by the correction term u − ˜ u is of Fuchsian type, which is a PDE
analogue of an ODE having a regular singular point. This has two consequences.
First, the parametrix needs to be constructed to very high order. This is done in
Section 3. Second, the contraction mapping argument revolves around a singular
variant of the linear wave equation.The basic estimates for this equation are
derived in Section 4 and then used to construct the solution in Section 5.
Working in one spatial dimension gives rise to a number of simplifications, both
mathematical and expository. We make particular use of the Minkowski-space ana-
logue of the Riemann mapping theorem (Proposition 2.2 below). This allows us
to choose coordinates (s,y) for the space-time region {t < σ(x)} that straighten
out the boundary while only deforming the d’Alembertian, ∂2
factor. That this is only possible in R1+1is a famous discovery of Liouville; specifi-
cally, the only (Euclidean or Minkowski) conformal mappings in higher dimensions
are M¨ obius transformations (see [5, §15]). As the notion of a conformal mapping
relative to a pseudo-Riemannian metric is discussed in relatively few differential
geometry textbooks ([5] is one exception), we develop the facts we require from
scratch in Section 2.
While M¨ obius transformations in Minkowski space have played a central role
in several fundamental developments in the theory of nonlinear wave equations
(spectacularly so at the hands of C. Morawetz or D. Christodoulou, for example),
we have not seen the greater generality available in the R1+1setting used previously
to aid in the analysis of this type of equation.
t−∂2
x, by a conformal
Acknowledgements. We are grateful to Mario Bonk for valuable discussions
about conformal mappings in Minkowski space.
The first author was partially supported by NSF grant DMS-1001531. The
second author was partially supported by the Sloan Foundation and NSF grant
DMS-0901166. This work was completed while the second author was a Harrington
Faculty Fellow at the University of Texas at Austin.
2. Existence and properties of conformal maps
In this section, we use a conformal mapping (relative to the Minkowski metric) to
reduce the problem of constructing solutions to (1.1) that blow up on a (sufficiently
smooth) space-like hypersurface to that of constructing solutions that blow up on

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4ROWAN KILLIP AND MONICA VIS ¸AN
the spatial axis; these latter solutions must solve a suitably modified nonlinear wave
equation.
Definition 2.1 (Conformal mapping). A C1diffeomorphism Φ : (s,y) ?→ (t,x)
between two open sets in Minkowski space R1+1is called conformal if there exists
a conformal factor λ(s,y) > 0 such that
?∂t
∂s
∂s
∂x
∂t
∂y
∂x
∂y
??−10
01
??∂t
∂s
∂t
∂y
∂x
∂s
∂x
∂y
?
= λ(s,y)
?−10
01
?
.(2.1)
Equivalently, the pull back of the Minkowski metric via Φ differs from the original
Minkowski metric by the scalar factor λ(s,y) > 0. The mapping Φ is a (Minkowski)
isometry at those points where λ(s,y) = 1.
From (2.1) it is clear that the null (or light-like) directions are unchanged by a
conformal mapping. As λ > 0, the notions of space- and time-like are preserved.
Thus there are four classes of conformal mappings depending on whether the spatial
and/or temporal orientations are preserved or inverted. When both are preserved,
the mapping is called proper (cf. [5, §6.2]).
It is easy to derive the Minkowski analogue of the Cauchy–Riemann equations
from (2.1). In the case of proper mappings, they read
∂t
∂s=∂x
∂y> 0 and
∂t
∂y=∂x
∂s.
In particular, (t,x) are wave coordinates, that is, tss− tyy= 0 and xss− xyy= 0
(in the sense of distributions). Using the general solution of the one-dimensional
wave equation, we deduce that all proper conformal mappings of Minkowski space
take the form
t =1
2
?f(y + s) − g(y − s)?
andx =1
2
?f(y + s) + g(y − s)?, (2.2)
where f and g are orientation-preserving C1diffeomorphisms of R.
maining three types of conformal mappings correspond to allowing f and/or g
to be orientation-reversing.) A more geometrical derivation of this fact (which
we learned from Mario Bonk) is as follows: As conformal mappings preserve the
null directions, the grid of light rays in (t,x) coordinates must be mapped onto
the corresponding grid in (s,y). Thus, a conformal mapping corresponds to an
(arbitrary and independent) change of variables in each of the null directions:
(x+t,x−t) = (f(y +s),g(y −s)), which is, of course, just another way of writing
(2.2).
(The re-
Proposition 2.2 (Existence of conformal maps). Let 2 ≤ k ∈ N∪{∞} and consider
a Ckuniformly space-like hypersurface Σ = {t = σ(x)}. Then there exists a Ck
conformal diffeomorphism from {s ≥ 0, y ∈ R} onto {t ≤ σ(x)}, which is an
isometry between the boundaries. Moreover, the conformal factor λ(s,y) is Ck−1
s,y.
Proof. The natural conformal mapping Φ : (s,y) ?→ (t,x) is not proper. We seek
instead one for which (s,y) ?→ Φ(−s,y) is proper, which, from the preceding dis-
cussion, must take the form
x + t = f(y − s) andx − t = g(y + s) (2.3)
with increasing functions f,g ∈ Ck(R).

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BLOWUP ON SMOOTH HYPERSURFACES FOR NLW5
As Σ is uniformly space-like, |σ′(x)| < 1 uniformly for x ∈ R. Thus, by the
implicit function theorem, there exists a Ckfunction h : R → R so that
t = σ(x)if and only ifx + t = h(x − t).
Moreover, ε2< h′(z) =
The condition that the conformal mapping sends {s = 0, y ∈ R} onto Σ reads
f(y) = h(g(y))
1+σ′(z/2+h(z)/2)
1−σ′(z/2+h(z)/2)< ε−2for some ε > 0.
for all y ∈ R,
while the condition that it is an isometry on the boundary is
f′(y)g′(y) = 1for all y ∈ R.
Therefore, g must solve the following ODE:
g′(y) =?h′(g(y))?−1/2
with, say,g(0) = 0.
(We take the positive square-root.) By Picard’s existence and uniqueness theorem,
this equation admits a unique global solution g ∈ Ckfor h ∈ Ckwith k ≥ 2. Thus
f = h ◦ g ∈ Ck, as well. Moreover g′(y) > ε and f′(y) > ε, so we are guaranteed
that both are diffeomorphisms onto the whole of R.
A computation shows that the conformal factor is given explicitly by
λ(s,y) := f′(y − s)g′(y + s) > 0,
s,y. It is now an easy matter to verify that the conformal mapping
we constructed satisfies the desired properties. This completes the proof of the
proposition.
which is clearly Ck−1
?
A direct consequence of Proposition 2.2 is the following
Corollary 2.3 (Reduction to blowup on lines). For any 2 ≤ k ∈ N∪{∞}, there is a
one-to-one correspondence between solutions to (1.1) that blowup on a Ckuniformly
space-like hypersurface in the sense of (1.3) and solutions to
vss− vyy−2(p+2)
p2
λ(s,y)|v|pv = 0(2.4)
that blow up on {s = 0, y ∈ R} in the sense that
lim
sց0s
2
pv(s,y) = 1 (2.5)
for each y ∈ R. Here, λ(s,y) denotes the conformal factor from Proposition 2.2.
Proof. Let Φ : (s,y) ?→ (t,x) be the conformal mapping from Proposition 2.2. A
straightforward computation shows that
∂2
s− ∂2
y= λ(s,y)(∂2
t− ∂2
x),
which means that v(s,y) = u◦Φ(s,y) solves (2.4) if and only if u(t,x) solves (1.1).
To prove the correspondence between the blowup behavior of v on the line {s =
0, y ∈ R} and that of u on the hypersurface Σ, we merely have to show that
σ(x) − t
s
lim
tրσ(x)
= −
?
1 − |σ′(x)|2= lim
sց0
σ(x) − t
s
, (2.6)
where (t,x) = Φ(s,y). In the former limit, x is held fixed; in the latter, y is fixed.
To verify these relations, we need to compute the derivative of Φ (and its inverse)
at the boundary.